2 research outputs found
Network conduciveness with application to the graph-coloring and independent-set optimization transitions
We introduce the notion of a network's conduciveness, a probabilistically
interpretable measure of how the network's structure allows it to be conducive
to roaming agents, in certain conditions, from one portion of the network to
another. We exemplify its use through an application to the two problems in
combinatorial optimization that, given an undirected graph, ask that its
so-called chromatic and independence numbers be found. Though NP-hard, when
solved on sequences of expanding random graphs there appear marked transitions
at which optimal solutions can be obtained substantially more easily than right
before them. We demonstrate that these phenomena can be understood by resorting
to the network that represents the solution space of the problems for each
graph and examining its conduciveness between the non-optimal solutions and the
optimal ones. At the said transitions, this network becomes strikingly more
conducive in the direction of the optimal solutions than it was just before
them, while at the same time becoming less conducive in the opposite direction.
We believe that, besides becoming useful also in other areas in which network
theory has a role to play, network conduciveness may become instrumental in
helping clarify further issues related to NP-hardness that remain poorly
understood
The conduciveness of CA-rule graphs
Given two subsets A and B of nodes in a directed graph, the conduciveness of
the graph from A to B is the ratio representing how many of the edges outgoing
from nodes in A are incoming to nodes in B. When the graph's nodes stand for
the possible solutions to certain problems of combinatorial optimization,
choosing its edges appropriately has been shown to lead to conduciveness
properties that provide useful insight into the performance of algorithms to
solve those problems. Here we study the conduciveness of CA-rule graphs, that
is, graphs whose node set is the set of all CA rules given a cell's number of
possible states and neighborhood size. We consider several different edge sets
interconnecting these nodes, both deterministic and random ones, and derive
analytical expressions for the resulting graph's conduciveness toward rules
having a fixed number of non-quiescent entries. We demonstrate that one of the
random edge sets, characterized by allowing nodes to be sparsely interconnected
across any Hamming distance between the corresponding rules, has the potential
of providing reasonable conduciveness toward the desired rules. We conjecture
that this may lie at the bottom of the best strategies known to date for
discovering complex rules to solve specific problems, all of an evolutionary
nature